Dynamical spectral rigidity and determination for billiard systems

台球系统的动态谱刚度及其测定

• 批准号: RGPIN-2022-04188
• 负责人: DeSimoi, Jacopo
• 金额: $ 2.26万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758284
• 关键词: Dynamical; spectral; rigidity; determination; billiard

The goal of inverse problems is the reconstruction of an object from a set of coarse observations. Inverse problems constitute a surprisingly broad class of problems with far-reaching applications in virtually any field of science: high energy physics (identifying particles created in scattering events), medicine (CT scans), astrophysics (detecting elements in the photosphere of stars which are billions of light-years away from us), image processing (de-noising and de-blurring of digital images), Big Data and Machine Learning. In this proposal I will describe the analysis of an inverse problem that can be purely formulated in the context of (classical) dynamical systems. Consider the trajectories of a particle that moves freely inside a planar domain and is subject to elastic reflections upon collisions with the boundary of the domain. We call periodic those trajectories that repeat themselves after a finite amount of time: such trajectories trace closed polygons inscribed in the domain. We call Length Spectrum of the domain the set of perimeters of all such polygons. We can then formulate the following inverse dynamical problem: Dynamical Spectral Determination: is it possible to identify the domain (modulo isometries) by the knowledge of its Length Spectrum? The above question, e.g. in the very natural class of smooth domains, is still wide open, and it is unarguably considered an extremely challenging problem. Sarnak conjectured that smooth domains are locally determined by their Laplace spectrum (a question that -quoting M. Kac- is often phrased as: "Can one hear the shape of a drum?"). Due to the tight connection between the Laplace (quantum) and dynamical (classical) spectral problems established by the Wave Trace formula, we find natural to study this conjecture in the dynamical setting. As a first step, we may consider a deformational problem: we say that a domain is dynamically spectrally rigid if all smooth deformations of the domain that preserve its Length Spectrum are necessarily isometries. In the past few years, together with my collaborators, we proved dynamical spectral rigidity for symmetric convex billiards close enough to disks. Also, we proved dynamical spectral determination for a class of (symmetric) analytic open dispersing billiards (such are systems whose dynamics is reminiscent of geodesic flow on manifolds with negative curvature). In the next several years, my research team and I will leverage on the breakthrough techniques that have been developed for the above results to move towards Sarnak's conjecture. On the one hand I will set to prove (local) spectral determination results for smooth convex billiards (possibly with symmetries); on the other hand I will push my work on hyperbolic billiards towards the problem of (local) spectral determination in the smooth category. I believe that these results will be attainable in this decade, and the Discovery Grant will play a major role in their development.

逆问题的目标是从一组粗略的观测数据中重建物体。逆问题构成了一个令人惊讶的广泛的问题，在几乎任何科学领域都有深远的应用：高能物理（识别散射事件中产生的粒子），医学（CT扫描），天体物理学（检测距离我们数十亿光年的恒星光球中的元素），图像处理（数字图像的去噪和去模糊），大数据和机器学习。在本提案中，我将描述对可以纯粹在（经典）动力系统背景下制定的逆问题的分析。考虑在平面区域内自由移动并在与区域边界碰撞时受到弹性反射的粒子的轨迹。我们称那些在有限时间内重复的轨迹为周期性的：这样的轨迹跟踪内接于域中的闭合多边形。我们称该域的长度谱为所有这些多边形的周长的集合。然后我们可以公式化下面的逆动力学问题：动力学谱确定：是否可能通过其长度谱的知识来识别域（模等距）？上面的问题，例如在非常自然的光滑域类中，仍然是一个很大的问题，它无疑被认为是一个非常具有挑战性的问题。Sarnak指出，光滑域是由它们的拉普拉斯谱局部确定的（引用M. Kac-经常被表述为：“一个人能听到鼓的形状吗？").由于波迹公式建立的拉普拉斯（量子）和动力学（经典）谱问题之间的紧密联系，我们发现在动力学背景下研究这个猜想是很自然的。作为第一步，我们可以考虑一个变形问题：我们说一个域是动态谱刚性的，如果所有保持其长度谱的域的光滑变形必然是等距的。在过去的几年里，我们和我的合作者一起，证明了足够接近圆盘的对称凸台球的动力学谱刚性。此外，我们证明了一类（对称）解析开放分散台球（这样的系统，其动力学是让人想起测地线流的负曲率流形）的动力学谱测定。在接下来的几年里，我和我的研究团队将利用为上述结果开发的突破性技术，向萨纳克猜想迈进。一方面，我将设置证明（本地）光谱测定结果光滑凸台球（可能与对称性）;另一方面，我将推动我的工作对双曲台球的问题（本地）光谱测定在光滑的类别。我相信，这些成果将在这十年内实现，而发现基金将在它们的发展中发挥重要作用。